Macaulay2 » Documentation
Packages » ResLengthThree :: ResLengthThree
next | previous | forward | backward | up | index | toc

ResLengthThree -- Computation of multiplicative structures on free resolutions of length three

Description

Let I be a homogeneous ideal contained in the irrelevant maximal ideal of a graded ring Q (obtained as a quotient of a polynomial ring). If the length of the minimal free resolution F of $R=Q/I$ is 3, then the resolution admits the structure of a differential graded algebra. The induced algebra structure on $A = Tor^Q(R,k)$ is unique and provides for a classification of such quotient rings. The package determines a multiplicative structure on the free resolution F as well as the unique induced structure on A and the class of the quotient R according to the classification scheme of Avramov, Kustin, and Miller.

Authors

Version

This documentation describes version 1.0 of ResLengthThree, released 3 December 2020.

Citation

If you have used this package in your research, please cite it as follows:

@misc{ResLengthThreeSource,
  title = {{ResLengthThree: Multiplication in free resolutions of length three. Version~1.0}},
  author = {Lars Winther Christensen and Luigi Ferraro and Francesca Gandini and Frank Moore and Oana Veliche},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}

Exports

  • Functions and commands
    • makeRes -- creates a resolution starting from three matrices
    • multTableOneOne -- the multiplication table for products of elements in degree one
    • multTableOneTwo -- the multiplication table for products of elements in degree one with elements in degree two
    • resLengthThreeAlg -- the minimal free resolution presented as a graded-commutative ring
    • resLengthThreeTorAlg -- the Tor algebra presented as a graded-commutative ring
    • resLengthThreeTorAlgClass -- the class (w.r.t. multiplication in homology) of an ideal
  • Methods
    • multTableOneOne(Ring) -- see multTableOneOne -- the multiplication table for products of elements in degree one
    • multTableOneTwo(Ring) -- see multTableOneTwo -- the multiplication table for products of elements in degree one with elements in degree two
    • resLengthThreeAlg(Complex) -- see resLengthThreeAlg -- the minimal free resolution presented as a graded-commutative ring
    • resLengthThreeAlg(Complex,List) -- see resLengthThreeAlg -- the minimal free resolution presented as a graded-commutative ring
    • resLengthThreeTorAlg(Complex) -- see resLengthThreeTorAlg -- the Tor algebra presented as a graded-commutative ring
    • resLengthThreeTorAlg(Complex,List) -- see resLengthThreeTorAlg -- the Tor algebra presented as a graded-commutative ring
    • resLengthThreeTorAlgClass(Complex) -- see resLengthThreeTorAlgClass -- the class (w.r.t. multiplication in homology) of an ideal
    • resLengthThreeTorAlgClass(Ideal) -- see resLengthThreeTorAlgClass -- the class (w.r.t. multiplication in homology) of an ideal
  • Symbols
    • Labels -- an optional argument for multTableOneOne and MultTableOneTwo determining whether to label rows and columns
    • Compact -- see multTableOneOne(...,Compact=>...) -- an optional argument for multTableOneOne that prints dots below the diagonal

For the programmer

The object ResLengthThree is a package, defined in ResLengthThree.m2.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/ResLengthThree.m2:386:0.